ON THE MOTION OF PENDULUMS.
133
importance. Making this simplification and substituting in (191) we get
where c=
If then J.0 be the initial and A the
final value of Ati we get from (192)
...(193).
Let now J.0 -h AJ.0 be what J.0 would become if, while the final arc A and the whole time t remained the same, the motion had been going on for an indefinite time before the epoch from which t is measured, in which case the superior limit in the integral involved in the expression for 6r would have been oo in place of t. Then
q = c f * jsin nt r n(t_ } dt^i dtf $ >
Jo ( Jo v&J
whence by subtracting, member from member, equation (193) from equation (194), we get
log
-o-o
-m nt fcos n (t _ 4 )
o I J^
which becomes after integration by parts
+
,
lo
c TT ft . ^ _ ^
= j / -- 2^ • cos nt - cos 2n£ cos nt -7-
Jt */t
8m2nt)[ sin w* ^[ ...... (195).
Now it is supposed to be very large : in Coulomb's experiments in fact 10 oscillations were observed, so that nt = lOyr. But when t is at all large the two integrals
r dt r - , dt
cos nt -77, sin ?££ —
7^ v^ Jt vt
can be expressed under the forms
— P sin nt -f Q cos w£, P cos 7^ -f Q sin n^, where
ll •'
i
^» 1 .3.5.